NOTE-133: Leitgeb & Ladyman — Criteria of identity and structuralism

Contribution

The PII debate over mathematical structuralism had turned on structures with non-trivial automorphisms whose places are still weakly discernible: the complex conjugates i and −i, the fermion singlet, and Black’s spheres, each linked by an irreflexive symmetric relation (Ladyman 2005; Saunders 2006). This note adds cases where even weak discernibility fails, taken from ordinary, non-pathological graph theory. It uses them to argue that a structuralist should take identity and distinctness of places as primitive and internal to the structure, with no criterion of identity behind them.

Key insight

If you follow what graph theorists actually count, the edgeless two-node graph G′ is one of exactly two unlabelled two-node graphs. Its two nodes are distinct, and nothing distinguishes them: not a property, not an irreflexive relation. “There are exactly two nodes” is part of what G′ is. So the structuralist slogan “there is no more to the nodes than the relations they bear to each other” survives if, and only if, identity and difference are counted among those relations.

Assumptions

• Methodological: the more closely a philosophy of mathematics matches how mathematicians “talk and reason”, the more prima facie plausible it is. The authors concede that philosophical claims “cannot actually be derived from descriptions of mathematical practice” (p. 2).
• Pattern (ante rem) structuralism: isomorphic structures are identical (p. 5). This is what makes “the” graph G′ unique once one such graph exists.
• Unlabelled graphs as the structuralist’s object: the set-theoretic reconstruction, which labels everything, “does not preserve” graph theorists’ practice (p. 2; footnote 4 quotes Urquhart).
• Weak discernibility: the PII in its weak form, where two objects are discernible if some irreflexive relation holds between them, is the principle being tested.
• The scope is mathematical objects. Whether physics has such cases is explicitly left open (pp. 1, 6).

Key results

• G (two nodes, one edge): symmetric, but the nodes are weakly discernible through the irreflexive relation “is connected by an edge to” (p. 3).
• G′ (two nodes, no edge): it has a non-trivial automorphism and no irreflexive relation between its nodes, yet graph theory counts it as a distinct unlabelled graph. Pólya enumeration theory exists to count such graphs (p. 4). So G′ violates even the weak PII.
• Generality: the point holds for every unlabelled graph with at least two isolated nodes (11 of the 156 unlabelled graphs on 6 nodes), and for every graph with two isomorphic connected components, G″ (p. 4).
• Thesis: the identity or difference of places is “not to be accounted for by anything else than the very graph itself” (p. 5). The identity relation is an integral component of a structure, as successor is of ℕ (p. 2).
• Against Button’s epistemic objection: the question is not how we know G′ has two nodes, but how we know such a graph exists. That is answered in standard ways: postulation, templates, and operations such as adding a node to the one-node graph or deleting the edge of G (p. 5).
• Against Button’s metaphysical objection: graph theorists quantify over nodes, count them and state identities of them, so they are “objects proper”. If metaphysicians call that a weak notion of object, “so be it” (p. 6).

Claims

id	claim	strength	support
C1	The edgeless two-node graph G′, and any graph with ≥2 isolated nodes or ≥2 isomorphic components, has distinct nodes that are not even weakly discernible.	strong	elementary graph theory, §2; checkable by hand
C2	Graph theorists treat such nodes as determinately distinct objects.	moderate	appeal to practice: Pólya enumeration, Tutte (fn. 2), Urquhart (fn. 4)
C3	Therefore the identity or difference of places is fixed by the structure alone, with no criterion of identity.	weak–moderate	informal argument from C1–C2 plus the methodological premise that fidelity to practice confers plausibility
C4	Counting identity as a structural relation adds nothing non-structural, whether identity is logical or mathematical.	moderate	dilemma argument, p. 4
C5	Irreducible identity facts do not commit structuralism to haecceitist Platonism, because permuting G′‘s nodes yields the same graph “even if the graph is labelled”.	weak	assertion for the labelled case, p. 5; in tension with fn. 3
C6	Dipert’s asymmetric-world-graph view rests on a mistaken demand that identity facts be grounded.	weak	brief argument, p. 6
C7	Whether physical objects can be objects in this “weak” sense is open, and bosons may either mark a disanalogy or show Saunders wrong.	—	posed as a question, p. 6

Concepts

• Weak discernibility: a and b are weakly discernible if some irreflexive relation holds between them (for example, “is connected by an edge to”). The PII in this form is what G′ violates.
• Unlabelled graph: a graph considered up to isomorphism, in which nodes are indistinguishable in isolation. Tutte calls it an “abstract graph”, that is, an isomorphism class.
• Labelled graph: an unlabelled graph plus names or numbers on its nodes. The authors compare this to Black’s “Castor and Pollux” (fn. 3).
• Primitive identity facts: facts about the identity or distinctness of places that are not grounded in any property or relation. Here they are counted as part of the structure.
• Two kinds of Platonism (p. 5): MacBride’s two-category Platonism, with objects and relations as separate categories of being, on which the authors are neutral; and Keranen’s haecceitist Platonism, which they reject.

Connections

The note takes sides in the 2004–2007 exchange among Keranen, MacBride, Ladyman (2005), Shapiro (2006, 2007) and Ketland (2006) on one side and Button (2006) on the other. It sides with Shapiro and Ketland against the demand for a criterion of identity. Ketland’s “dumb-bell” structure (fn. 1) and Button’s own examples are acknowledged as similar cases that were given different morals. The physics side is Saunders (2006), who argued that fermions are weakly discernible and that bosons are not objects.

Among held works, Chakravartty’s chapter (LIT-196, read as x6) takes up exactly this graph-theoretic line. It targets Dipert and Ladyman’s 2007 asymmetric labelled graph, not this note, and argues that graph labels identify locations in a structure, not the identity of whatever occupies them in the concrete world. So the mathematics/physics disanalogy this note leaves open is the point on which Chakravartty presses. Quasi-set theory (LIT-107) goes the other way for quantum items: it drops identity rather than taking it as primitive. The SEP Structural Realism entry (LIT-045) places weak discernibility within the OSR debate. The relation-priority views in LIT-173 (Morganti) and LIT-152 (Floridi) inherit the question of whether primitive identity of places is available for physical structure.

Bearing on metaphysics: the note defends a metaphysical thesis about mathematical objects. Their numerical diversity is primitive and structure-internal: it is not grounded in properties, relations or haecceities. It also argues that metaphysicians’ standards of objecthood should give way to mathematical practice. The tag is justified. identity is the right primary topic.

Bearing on the record

It supports any THEORY-level claim in this record that structuralism about mathematics can accept indiscernible but distinct places. It does not support the corresponding claim for physics: the authors explicitly withhold it. It carries nothing for ML practice, and nothing here belongs in the Anthology of the SOTA.

Note of 2026-10-16: THEORY-032 now cites LIT-154 as a source.

Limitations

• The central inference, from what graph theorists do to what places in a structure are, rests on a methodological premise that the authors themselves say cannot be derived from practice.
• A critic who wants a criterion of identity can reply that the practice is well served by the set-theoretic (labelled) reconstruction, and that “unlabelled” talk is a façon de parler. The note answers this with Urquhart’s remark (fn. 4), not with an argument.
• The claim that permutation leaves even a labelled G′ unchanged (p. 5) is asserted, not argued, and it is hard to square with fn. 3.
• The epistemological reply to Button explicitly sets aside the general epistemic worries about pattern structuralism (what “coherent” means, how templates yield knowledge).
• It is a discussion note of about 7 pp., with no formal apparatus beyond the examples.

Open questions

• Are there physical systems with non-trivial automorphisms and no irreflexive relations between their elements? Bosons are the candidate. Is the answer an empirical matter, as the Carnap railway example suggests?
• If identity is part of a structure, does ante rem structuralism need MacBride’s two-category Platonism after all? The authors stay neutral.
• Does the argument transfer to ontic structural realism about physical structure? Chakravartty (LIT-196) argues that it does not, absent Pythagoreanism.

Corrections to the seeded skim

• The dossier says “The identity relation belongs to the structure, like successor in the natural numbers”. That is the thesis (§1, p. 2), but the dossier leaves out half of the §2 defence (p. 4): identity is structural whether it is taken as a logical relation, in which case it is as structural as existence, or as a genuinely mathematical one, in which case the structuralist should count it part of the structure like any other mathematical relation. The note rests on this dilemma.
• The dossier leaves out G″ (p. 4): graphs with two isomorphic connected components and no edge between them. This generalises the point from isolated nodes to whole indistinguishable components, and it is why the authors deny that G′ is “exceptional or pathological”. The dossier also leaves out the 11-of-156 figure for 6-node graphs.
• The dossier says the authors “stay neutral on the weaker two-category Platonism”. More exactly (p. 5), they deny that irreducible identity facts commit them to it, and stay neutral only on whether that weaker Platonism is true, because it seems compatible with ante rem structuralism. They reject Keranen’s haecceitist Platonism outright: permuting G′‘s nodes yields the same graph, “even if the graph is labelled”. That last clause is asserted without argument, and it sits awkwardly with their own footnote 3, which says labels make nodes distinguishable.
• The closing is richer than the dossier says (p. 6). Besides the fermion/boson disanalogy, it rejects Dipert’s (1997) claim that the world is an asymmetric graph, on the ground that his supervenience requirement rests on the “mistaken claim” that identity facts must be grounded. It also cites Carnap’s Aufbau railway-network example (§14) for a world with non-trivial automorphisms, which it leaves as an open empirical question.
• The byline is Leitgeb and Ladyman, and both are at Bristol in the manuscript. The batch’s “Leitgib” is a misspelling, as the dossier already notes.
• metaphysics tag: justified. The note defends a thesis about what there is and what grounds numerical diversity: primitive, structure-internal identity facts, and no haecceities. §3 is titled “Mathematics versus Metaphysics” and replies to MacBride’s and Button’s metaphysical objections. identity rightly leads. mathematics is only a near fit, because the paper is philosophy of mathematics, not “mathematics read for itself”. The vocabulary has no philosophy-of-mathematics topic, and a proposal for one could cite this note.